611 



T applied to v (the name has been introduced by Pinoheri>r) are 

 known ; a development of Tio in the point w = v, or in the 

 vicinity of lo ^z v, as we might say, while the special development, 

 according to powers of Did, might be called one in the point y/;=l. 

 (Cf. the general development of a fnnction f {x) in the point x =: a 

 and the particular one in the point ,v = 0). 

 We again make the following suppositions: 



1. T is normal such that the N.F. is a circle («) with centre a'„, 

 while the functions of the F.F. belong to a circle (a), concentric 

 with («). 



2. The series F belonging to T is complete in («) with corre- 

 sponding domain {[f). 



It will further be easy in the future to suppose that as a pair 

 of associated fields of T each such pair of fields of P comes into 

 consideration, which involves that for {a) may be taken any circle 

 greater than 0?) (see N°. 12, 2"'^' communication). Possibly (^) itself 

 cannot be taken as N.F.F. together with («) as N.F.O. (N". 13); but 

 in any case we have now according to the functional theorem of 

 Mac-Laurin, Tu = Pii, for all functions belonging to {^), because, 

 if u belongs to (^), it also belongs to a somewhat larger circle. 

 According to the final observation in N". 15 (3''' communication), 

 if T were not yet defined for all these functions, we might extend 

 the F. F. of T over them by simply writing for these functions 

 7u ■= Pu. And if T should be defined a pi-iori for a certain part 

 of the functions belonging to (/?), such that we had for them 

 Tu=\=Pu, T would not be continuous in the total field, and in 

 that case we should not retain this T in the considerations we are 

 concerned with here, but put another operation in its place, which 

 for the functions mentioned is identical with P. 



If therefore ir> be a function belonging to (|3), we shall have in 

 the domain («) 



1 w =: rxv ^E > " . 







Now we have, since iv=^vu, according to the formula of Leibniz 

 consequently 



00 



7'.?/'==:\w — (üm(") + Wjü'm("— 1) -|- . . . -}- uWm). 

 



In each member of this last series, from and after the one 

 for which n = m, occurs a term with ft'"'; taking these terms 



